Abstract : This mini-symposium brings together leading experts in the field of systems of PDEs arising in the context of interacting particles. Steric effects and interactions between members of opposite or the same species typically lead to systems of nonlocal and cross-diffusion type. The interplay of degenerate parabolicity and nonlocalities leads to a myriad of interesting emergent behaviours including pattern formation and phase separation. At the same time, these systems pose a variety of challenging analytical mathematical problems including the dramatic loss of regularity at the onset of phase separation. Thus, new analytical techniques and reliable numerical methods are needed.
Organizer(s) : Jan-Frederik Pietschmann, Markus Schmidtchen, Havva Yoldaş
00134 (1/3) : 2C @D407 [Chair: Markus Schmidtchen]
[05128] Evolution Equations for Interacting Species: Applications and Analysis
Format : Online Talk on Zoom
Author(s) :
Jan-Frederik Pietschmann (University of Augsburg)
Markus Schmidtchen (Technische Universität Dresden)
Havva Yoldaş (Delft University of Technology)
Abstract : This talks provides an overview of the mini-symposium's topic of systems of PDEs arising in the context of interacting particles. Steric effects and interactions between members of opposite or the same species typically lead to systems of nonlocal and cross-diffusion type. The interplay of degenerate parabolicity and nonlocalities leads to a myriad of interesting emergent behaviours including pattern formation and phase separation. At the same time, these systems pose a variety of challenging analytical mathematical problems including the dramatic loss of regularity at the onset of phase separation. Thus, new analytical techniques and reliable numerical methods are needed.
[05084] A Keller-Segel type approximation to a cell population dynamics model
Format : Talk at Waseda University
Author(s) :
Hideki Murakawa (Ryukoku University)
Abstract : We deal with a cell population dynamics model with nonlocal advection term. Approximating non-local advection as a local problem can be useful for analysis and numerical analysis of the problem. In this talk, we present a Keller-Segel type approximation to the cell population dynamics model. The approximation consists only of local terms. We discuss convergence of the approximation and introduce some applications. This is a joint work with Yoshitaro Tanaka.
[04550] Convergence of position-based dynamics for first-order particle systems with volume exclusion
Format : Talk at Waseda University
Author(s) :
Steffen Plunder (Kyoto University, ASHBi)
Sara Merino-Aceituno (University of Vienna)
Abstract : To simulate first-order particle systems with volume exclusion, we adapted the position based dynamics (PBD) method from computer graphics. PBD is a simple, fast and explicit time-stepping method which is unconditionally stable.
Our contribution is the first convergence proof for PBD for first-order systems. Our proof uses the theory of differential inclusions on uniformly prox-regular sets and a new error estimate for alternating projections.
We successfully applied the method in various applications in developmental biology.
[04803] Graph-to-local limit for the nonlocal interaction equation
Format : Talk at Waseda University
Author(s) :
Georg Heinze (University of Augsburg)
Abstract : In this talk I will discuss a proof of existence of solutions for the nonlocal interaction equation in Euclidean space using a graph-based nonlocal approximation. The graph equations are induced by an upwind interpolation, which leads to non-symmetric graph gradient structures that nevertheless converge to a symmetric Wasserstein-type local gradient structure. Furthermore, the flexibility of our graph model allows us to introduce a tensor to modify the geometry underlying the limiting model.
[05193] A model for territorial dynamics: from particle to continuum
Format : Online Talk on Zoom
Author(s) :
Alethea Barbaro (TU Delft)
Abdulaziz Alsenafi (Kuwait University)
Abstract : Many species, including our own, demonstrate territoriality, with individuals or groups marking their territories either chemically or visually. Here, we present an agent-based lattice model for territorial development. In this model, there are several groups; agents from each group put down that group’s territorial markings as they move on the lattice. Agents move away from areas with territorial markings which do not belong to their own group. The model was motivated by gangs expressing territoriality through graffiti markings, though the model itself could be applicable in any chemo-repellent situation. We show that this model undergoes a phase transition between well-mixed dynamics and the formation of distinct territories as parameters are varied. We formally derive a system of coupled convection-diffusion equations from this model. The system is cross-diffusive due to the avoidance of other groups’ markings. Using the PDE system, we pinpoint the critical value for the phase transition.
[04662] A variational approach for an existence result for a cross-diffusion model
Format : Talk at Waseda University
Author(s) :
Havva Yoldaş (Delft University of Technology)
Filippo Santambrogio (Université Claude Bernard - Lyon 1)
Romain Ducasse (Université Paris Cité)
Abstract : In this talk, we look at a cross-diffusion system consisting of two Fokker-Planck equations where the gradient of the density for each species acts as a potential for the other one. The system is the gradient flow for the Wasserstein distance of a functional which is not lower semi-continuous, and the system is not well-posed. We compute the convexification of the integral and provide an existence proof in a suitable sense for the gradient flow of the corresponding relaxed functional.
[04745] A degenerate cross-diffusion system as the inviscid limit of a nonlocal tissue growth model
Format : Talk at Waseda University
Author(s) :
Tomasz Dębiec (University of Warsaw)
Abstract : In recent years, there has been a spike in interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this talk, we want to address this dearth of results in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type).
[04637] Proposing a Finite Volume Method for a Kinetic Model for Interacting Species
Format : Talk at Waseda University
Author(s) :
Julia Ines Mareike Hauser (TU Dresden)
Abstract : We consider a system of two kinetic equations coupled by non-local interaction terms which are used to describe systems of indistinguishable agents such as flocks of birds.
In this talk we propose an upwind finite volume method for this model. The method is constructed in such that mass is preserved and positivity is maintained. We show the convergence of the method and we provide explicit error estimates. Finally, we underline our theoretical results with simulations.
[05038] Mean-field convergence in L^2-norm for a diffusion model with aggregation
Format : Online Talk on Zoom
Author(s) :
Alexandra Holzinger (TU Wien )
Abstract : Aggregation effects appear in many applications arising from biology and physics which makes it interesting to study this phenomena also in mean-field settings. It is well-known that a class of local diffusion-aggregation equations can bederived by using classical mean-field limits. In this talk I will explain the benefits we get by showing a result in L^2-norm and how this is connected to fluctuations around the mean-field limit.
[04613] Macroscopic limits of kinetic equations for the switch in cell migration via binary interactions
Format : Talk at Waseda University
Author(s) :
Gissell Estrada-Rodriguez (University of OxfordU)
Abstract : Motivated by experimental results on the immune response to cancer, we considered a system of particles, I, in an
inactive state, where they follow a nonlocal (Levy) movement. After a collision with particles in population D, they
change to an active state, A, resulting in a more localised (Brownian) movement. Activation is described via binary
interactions between I and D. Moreover, cell motion is represented as a velocity-jump process, with the running time of I following a long-tailed distribution, which is consistent with a Levy walk, and the running time of A following a Poisson
distribution, which corresponds to Brownian motion. We formally show that the macroscopic limit of the model
comprises a coupled system of balance equations for the one-particle distribution functions of populations I, D and A. The modelling approach presented here and its possible generalisations are expected to find applications in the study of the immune response to cancer and in other biological contexts in which switch from non-local to localised migration patterns occurs.
[04647] Towards a new mathematical model of the visual cycle
Abstract : The visual cycle is a fundamental bio-chemical process allowing photoreceptors to convert light into electrical signals and return to the dark state. I will present a new mathematical model involving coupled ODEs and PDEs for the kinetics of retinal photo-sensitive molecules after light exposure. This reaction-diffusion-type model provides a first step in the study of the accumulation of toxic by-products in the eye in connection with retinal diseases such as age-related macular degeneration.
[04418] Structured Model for the Size-spectrum Evolution in Aquatic Ecosystems
Format : Talk at Waseda University
Author(s) :
Laura Kanzler (CEREMADE - Université Paris-Dauphine)
Abstract : Trophic interactions between animals in the ocean were matter of interest since the 60’, where it was quickly discovered that the individuals’ body size acts as ’master trait’ in food webs of animals, giving rise to emergent distributions of biomass, abundance and production of organisms.
We propose and investigate a deterministic jump-growth model of Boltzmann type, aiming to capture this emergence phenomenon in aquatic ecosystems. The equation of interest is derived from individual based dynamics governed by a stochastic process. Following the observation of the body mass being the crucial trait in these dynamics it is based on the assumption that binary interactions between individuals in the ecosystem take place: A predator feeding on a prey, which then results in growth of the predator with assimilating a certain (usually very small) amount of its prey’s mass as well as plankton production. Analytical results in various parameter regimes are discussed and numerical simulations underlying these observations are given.
